Use the exponential distribution to model Poisson processes, in which events occur continuously and independently at a constant rate.

This distribution has a wide range of applications, including reliability analysis of products and systems, queuing theory, and Markov chains.

For example, the exponential distribution can be used to model:

- How long it takes for electronic components to fail
- The time interval between customers' arrivals at a terminal
- Service time for customers waiting in line
- The time until default on a payment (credit risk modeling)
- The distance between mutations on a DNA strand
- Time until a radioactive nucleus decays

The 1-parameter exponential distribution is defined by its scale parameter.The 2-parameter exponential distribution is defined by its scale and threshold parameters.The threshold parameter, λ, if positive, shifts the distribution by a distance λ to the right.

For example, you are interested in studying the failure of a system with λ = 5. This means that the failures start to occur only after 5 hours of operation and cannot occur before.

An important property of the exponential distribution is that it is memoryless. The chance of an event occurring does not depend on elapsed time. Therefore, the occurrence rate remains constant as the process or system does not "remember" its past. For example, the memoryless property indicates that the remaining life of a component is independent of its current age. A system that had wear and tear and thus becomes more likely to fail later in its life is not memoryless.